I think the point is that mathematical reasoning is inherently self-correcting in this sense, and that this corrective force is intentionistic and Lamarckian - it is being corrected toward a mathematical argument which one thinks of as a timeless perfect Form (because come on, are there really any mathematicians who don't, secretly, believe in the Platonic realism of mathematics?), and not just away from an argument that's flawed.
An incorrect theory can appear to be supported by experimental results (with probability going to 0 as the sample size goes to \infty), and if you have the finite set of experimental results pointing to the wrong conclusion, then no amount of mind-internal examination of those results can correct the error (if it could, your theory would not be predictive; conservation of probability, you all know that). But mind-internal examination of a mathematical argument, without any further entangling (so no new information, in the Bayesian sense, about the outside world; only new information about the world inside your head), can discover the error, and once it has done so, it is typically a mechanical process to verify that the error is indeed an error and that the correction has indeed corrected that error.
This remains true if the error is an error of omission (We haven't found the proof that T, so we don't know that T, but in fact there is a proof of T).
So you're not getting new bits from observed reality, yet you're making new discoveries and overthrowing past mistakes. The bits are coming from the processing; your ignorance has decreased by computation without the acquisition of bits by entangling with the world. That's why deductive knowledge is categorically different, and why errors in logical reasoning are not a problem with the idea of logical reasoning itself, nor do they exclude a mathematical statement from being unconditionally true. They just exclude the possibility of unconditional knowledge.
Can you conceive of a world in which, say, ⋀∅ is false? It's certainly a lot harder than conceiving of a world in which earplugs obey "2+2=3"-arithmetic, but is your belief that ⋀∅ unconditional? What is the absolutely most fundamentally obvious tautology you can think of, and is your belief in it unconditional? If not, what kind of evidence could there be against it? It seems to me that ¬⋀∅ would require "there exists a false proposition which is an element of the empty set"; in order to make an error there I'd have to have made an error in looking up a definition, in which case I'm not really talking about ⋀∅ when I assert its truth; nonetheless the thing I am talking about is a tautological truth and so one still exists (I may have gained or lost a 'box', here, in which case things don't work).
My mind is beginning to melt and I think I've drifted off topic a little. I should go to bed. (Sorry for rambling)
I guess there are my beliefs-which-predict-my-expectations and my aliefs-which-still-weird-me-out. In the sense of beliefs which predict my expectation, I would say the following about mathematics: as far as logic is concerned, I have seen (with my eyes, connected to neurons, and so on) the proof that from P&-P anything follows, and since I do want to distinguish "truth" from "falsehood", I view it as (unless I made a mistake in the proof of P&-P->Q, which I view as highly unlikely -- an easy million-to-one against) as false....
In “What is Evidence?” I wrote:1
Cihan Baran replied:2
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations,” and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 - 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx - xx = xx.
I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.
How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.3
What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.
There was a time when I had no idea that 2 + 2 = 4. I did not arrive at this new belief by random processes—then there would have been no particular reason for my brain to end up storing “2 + 2 = 4” instead of “2 + 2 = 7.” The fact that my brain stores an answer surprisingly similar to what happens when I lay down two earplugs alongside two earplugs, calls forth an explanation of what entanglement produces this strange mirroring of mind and reality.
There’s really only two possibilities, for a belief of fact—either the belief got there via a mind-reality entangling process, or not. If not, the belief can’t be correct except by coincidence. For beliefs with the slightest shred of internal complexity (requiring a computer program of more than 10 bits to simulate), the space of possibilities is large enough that coincidence vanishes.4
Unconditional facts are not the same as unconditional beliefs. If entangled evidence convinces me that a fact is unconditional, this doesn’t mean I always believed in the fact without need of entangled evidence.
I believe that 2 + 2 = 4, and I find it quite easy to conceive of a situation which would convince me that 2 + 2 = 3. Namely, the same sort of situation that currently convinces me that 2 + 2 = 4. Thus I do not fear that I am a victim of blind faith.5
1See Map and Territory.
2Comment: http://lesswrong.com/lw/jl/what_is_evidence/f7h.
3See “Your Strength as a Rationalist” in Map and Territory.
4For more on belief formation and beliefs of fact, see “Feeling Rational” and “What Is Evidence?” in Map and Territory. For more on belief complexity, see “Occam’s Razor” in the same volume.
5If there are any Christians reading this who know Bayes’s Theorem, might I inquire of you what situation would convince you of the truth of Islam? Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity: We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.
Or is there more to it than that? If so, what situation would convince you of Islam, or at least, non-Christianity? And how confident are you that the general kinds of evidence and reasoning you appeal to would have been enough to dissuade you of your religion if you had been raised a Muslim?